Mathematica says
In[49]:= Remove[F, k, l, a, b, r]
Solve[(3/4) ((F + k (l - a))/(b a^(3/2) r^(1/2))) == 1, a]
Out[50]= {{a -> (3 k^2)/(
16 b^2 r) - (-((81 k^4)/(b^4 r^2)) + (864 k (F + k l))/(
b^2 r))/(144 3^(
1/3) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(b^4 r^2) - (144 k^4 l)/(
b^4 r^2) + (384 F^2)/(b^2 r) + (768 F k l)/(b^2 r) + (
384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)) + 1/16 3^(
1/3) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(b^4 r^2) - (144 k^4 l)/(
b^4 r^2) + (384 F^2)/(b^2 r) + (768 F k l)/(b^2 r) + (
384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)}, {a -> (3 k^2)/(
16 b^2 r) + ((1 + I Sqrt[3]) (-((81 k^4)/(b^4 r^2)) + (
864 k (F + k l))/(b^2 r)))/(288 3^(
1/3) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(b^4 r^2) - (144 k^4 l)/(
b^4 r^2) + (384 F^2)/(b^2 r) + (768 F k l)/(b^2 r) + (
384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)) - 1/32 3^(
1/3) (1 - I Sqrt[3]) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(
b^4 r^2) - (144 k^4 l)/(b^4 r^2) + (384 F^2)/(b^2 r) + (
768 F k l)/(b^2 r) + (384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)}, {a -> (3 k^2)/(
16 b^2 r) + ((1 - I Sqrt[3]) (-((81 k^4)/(b^4 r^2)) + (
864 k (F + k l))/(b^2 r)))/(288 3^(
1/3) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(b^4 r^2) - (144 k^4 l)/(
b^4 r^2) + (384 F^2)/(b^2 r) + (768 F k l)/(b^2 r) + (
384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)) - 1/32 3^(
1/3) (1 + I Sqrt[3]) ((9 k^6)/(b^6 r^3) - (144 F k^3)/(
b^4 r^2) - (144 k^4 l)/(b^4 r^2) + (384 F^2)/(b^2 r) + (
768 F k l)/(b^2 r) + (384 k^2 l^2)/(b^2 r) + (
64 Sqrt[3]
Sqrt[-F^3 k^3 - 3 F^2 k^4 l - 3 F k^5 l^2 - k^6 l^3 +
12 b^2 F^4 r + 48 b^2 F^3 k l r + 72 b^2 F^2 k^2 l^2 r +
48 b^2 F k^3 l^3 r + 12 b^2 k^4 l^4 r])/(b^3 r^(3/2)))^(
1/3)}}